The riots that have rampaged across Greece may have many causes, but one that is rarely mentioned is the fracturing of the Greek left into George Papandreou's traditional socialist party, Pasok, and an increasingly radicalised faction that refuses all accommodation with either the European Union or modern economics. To varying degrees, this divide is paralysing socialist parties across Europe.

That the traditional left is so inert in the midst of today's economic crisis is more than strange. Instead of thriving on renewed doubts about capitalism, Europe's socialist parties have failed to make any serious political inroads. In countries where they hold power, such as Spain, they are now very unpopular.

Where they are in opposition, as in France and Italy, they are in disarray – as is Germany's Social Democratic party (SDP), despite their being part of the ruling grand coalition. Even Sweden's out-of-power Socialists, the country's dominant party for a century, have failed to capitalise on the crisis. The United Kingdom may be the exception, although the pro-market Labour party shaped by Tony Blair may not count as a party of the left anymore.

European socialists have failed to address the crisis cogently because of their internal divisions. Born anti-capitalist, these parties all (to greater and lesser degrees) came to accept the free market as the foundation of the economy. Moreover, since 1991 and the collapse of the Soviet system, the left has lacked a clear model with which to oppose capitalism.

But, despite paying lip service to the market, the European left remains torn by an inner contradiction between its anti-capitalist origins and its recent conversion to free-market economics. Is the present crisis a crisis of capitalism or just a phase of it? This controversy keeps left-wing intellectuals, pundits, and politicians busy on television talk shows and in café debates across Europe.

As a result, a struggle for power has erupted. In France and Germany, a new far left – composed of Trotskyites, communists, and anarchists – is rising from the ashes to become a political force again. These rejuvenated ghosts take the form of Oskar Lafontaine's Left party in Germany, as well as various revolutionary movements in France; one of them has just named itself the Anti-Capitalist party. Its leader, a onetime postman, says that in the present circumstances, he is part of a "resistance", a word resonant of the anti-fascist struggles of the Hitler era. The actual alternative to capitalism that this far left seeks is anyone's guess.

In the face of this new radicalism, which is attracting some traditional socialists, what are the more established socialist leaders to do? When they bend towards the Trotskyites, they lose "bourgeois" supporters; when they seek the middle ground, like the SDP in Germany, the Left party grows. As a consequence of this dilemma, socialist parties across Europe seem paralysed.

And they are. Indeed, it is hard to find any persuasive analysis of today's crisis from the left beyond anti-capitalist slogans. The socialists blame greedy financiers, but who doesn't? In terms of remedies, the socialists do not offer anything more than the Keynesian solutions that are now being proposed by the right.

Since George W Bush showed the way towards bank nationalisation, vast public spending, industrial bailouts, and budget deficits, the socialists have been left without wiggle room. French president Nicolas Sarkozy tries to rekindle growth through the protectionist defence of "national industries" and huge investments in public infrastructure, so what more can socialists ask for? Moreover, many socialists fear that excessive public spending may trigger inflation, and that their core constituencies will become its first victims.

When the right has turned statist and Keynesian, the free market's true believers are marginalised, and old-style anti-capitalism seems archaic, one wonders what socialism in Europe can possibly mean?

The future of European socialism is also hampered, strangely, by the EU. To build socialism in one country is impossible nowadays because all of Europe's economies are now interdependent. The last leader to try go-it-alone socialism, French president François Mitterrand in 1981, surrendered to the European institutions in 1983.

These institutions, based on free trade, competition, limited budget deficits, and sound money, are fundamentally pro-market; there is little leeway within them for doctrinaire socialism. This is why the far left is anti-European.

European socialists are also finding it hard to distinguish themselves in foreign affairs. They used to be reflexively pro-human rights, much more so than conservative parties. But ever since Bush included these ideas as part of his democracy-promotion campaigns, European socialists have become more wary of them.

Moreover, without the Soviet Union, European socialists have few foreign causes to take to heart: few understand Putin's Russia, and today's totalitarian-capitalist China is too far and too strange. And, since Barack Obama's election, anti-Americanism is no longer a viable way to garner support. The good old days when Trotskyites and socialists found common ground in bashing the United States are over.

The ideological weakness and division of the left will not, of course, exclude them from power. Leaders can cling to office, as José Zapatero is doing in Spain and Gordon Brown is doing in the UK. The left may ultimately win general elections elsewhere if the new Keynesian right proves unable to end the crisis. But whether in opposition or in power, the socialists have no distinct agenda.

The lesson from Greece, however, is that what European socialists should fear most is the far left's taste and talent for disruption. For the hollowing out of socialism has a consequence. To paraphrase Marx, a spectre is haunting Europe – the spectre of chaos.